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A generalized transversality in global analysis. (English) Zbl 1152.46061

Summary: Let \(M\) and \(N\) be \(C^r\) Banach manifolds with \(r \geq 1\). Let \(P\) be a submanifold of \(N\) and \(f : M \rightarrow N\) a \(C^r\) map. This paper extends the well-known transversality \(f\pitchfork P \bmod N\) to the tangent map \(T_xf\) with a sharper singularity by using a new characteristic of the continuity of generalized inverses of linear operators in Banach spaces under small perturbations. We introduce a concept of generalized transversality, written as \(f\pitchfork_G P \bmod N\). We show that if \(f\pitchfork P\bmod N\), then \(f\pitchfork_G P \bmod N\), but the converse is false in general. Then, Thom’s famous result is expanded into a generalized transversality theorem: if \(f\pitchfork_G P \bmod N\), then the preimage \(S = f^{-1}(P)\) is a submanifold of \(M\) with the tangent space \(T_xS = (T_xf)^{-1}(T_{f(x)}P)\) for any \(x \in S\). As a consequence, when \(P=\{y\}\) is a single point set, \(f\pitchfork_G P\text{ mod } N\) if and only if \(y\) is a generalized regular value of \(f\). Finally, we give an equivalent geometric description of generalized transversality without the aid of charts.

MSC:

46T05 Infinite-dimensional manifolds
47A55 Perturbation theory of linear operators
58C15 Implicit function theorems; global Newton methods on manifolds
58K99 Theory of singularities and catastrophe theory
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